77 G1.02: Intercepts & Example 3
y-intercept
In algebra class, we learned that the y-intercept of a graph is the value on the y-axis when x = 0. That is easy to find using the formula for the line, because we only have to plug in x = 0 and find y.
Since [latex]F=1.8\cdot{C}+32[/latex], when we plug in C = 0, then [latex]F=1.8\cdot{C}+32=1.8\cdot0+32=32[/latex]
When we look at the graph for this formula, go to C = 0, and find the point on the graph above that, the y-value appears to be about a little bit above 30. To find a more precise estimate from the graph, we’d need a bigger graph. This is consistent with the value of 32 that we found algebraically. Understanding both of these provides a way of checking your work on either of them.
The formula for the line
Notice that both the slope and the y-intercept appear in the formula for the line. In algebra class, you learned that
- Any equation in which x and y appear only to the first power is a linear equation.
- The equation of a line can be written in a variety of ways, but if you take any of those and “solve for y” you’ll get a formula like [latex]y=mx+b[/latex], where y is the output variable, x is the input variable, and m and b are numbers. Then the slope is the coefficient of the x-variable and the y-intercept is the constant. So m is the slope and b is the y-intercept.
For the temperature example, since F is the output value and C is the input value, and [latex]F=1.8\cdot{C}+32[/latex], then we see from the formula that the slope is 1.8 and the y-intercept is 32.
Interpret the slope and intercept
- The y-intercept is the value for y when
- When the x-value increases by 1 unit, then the y-value increases by the value of the slope.
So, for the temperature problem, we found that if the temperature was 0° C, then it is 32° F. And we also found that, if the temperature C increases by 1° C, then the temperature F increases by 1.8° F.
Example 1. For this formula: [latex]R=-6.3+8.2\cdot{d}[/latex], tell
- Which variable is the output variable?
- Which variable is the input variable?
- Is it linear relationship?
- If it is a linear relationship, what is the slope?
- If it is a linear relationship, what is the y-intercept?
Show Answer
- The output variable is R.
- the input variable is d.
- yes, it is a linear relationship because both variables appear only to the first power.
- the slope is 8.2 because that is the number multiplied by the input variable.
- the y-intercept is -6.3.
Example 2. For this formula: [latex]R=-6.3+8.2\cdot{d}[/latex], find three points that fit this and use them to sketch a graph.
Show Answer
To find a point that fits this relationship, choose any value for the input variable. (Generally we choose a value that is easy to compute with and to graph, such as a small whole number.) Let’s begin by choosing [latex]d=2[/latex]. Then compute the output value.
[latex]\begin{align}&R=-6.3+8.2\cdot{d}\\&R=-6.3+8.2\cdot(2)\\&R=-6.3+16.4\\&R=10.1\\\end{align}[/latex]
Now choose two other values for d and use them to compute the output values. In the interests of saving space, we will omit the details.
Choose [latex]d=5[/latex] and find that [latex]R=-6.3+8.2\cdot5=34.7[/latex]
Choose [latex]d=7[/latex] and find that [latex]R=-6.3+8.2\cdot7=51.1[/latex]
| Graph with unconnected points | Graph with points connected by a line |
Example 3. Determine whether these three points lie on a straight line by computing slopes: (3,5) and (1,4) and (4,6).
Show Answer
Find the slope of the line through (3,5) and (1,4). [latex]m=\frac{rise}{run}=\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}=\frac{5-4}{3-1}=\frac{1}{2}=0.5[/latex]
Find the slope of the line through (3,5) and (4,6). [latex]m=\frac{rise}{run}=\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}=\frac{5-6}{3-4}=\frac{-1}{-1}=1.0[/latex]
Since these two slopes aren’t equal, these three points are not on a straight line.
Check: Check this by graphing the three points to see if they appear to lie on a straight line. In the following graph, we have drawn the line between the points and it is clear that it is not the same straight line between all three points.